Question

How many photons per second must strike a surface of area \(10.0 \mathrm{~m}^{2}\) to produce a force of \(0.100 \mathrm{~N}\) on the surface, if the photons are monochromatic light of wavelength \(600 . \mathrm{nm}\) ? Assume the photons are absorbed.

Short answer

Solution: 1. Find the energy of a single photon: \(E = \dfrac{hc}{\lambda}\) 2. Find the momentum of a single photon: \(p = \dfrac{E}{c}\) 3. Calculate the force per photon: \(\Delta p_\text{(photon)} = p_\text{(photon)} \times \dfrac{dN}{dt} \) 4. Calculate the number of photons per second: \(\dfrac{dN}{dt} = \dfrac{F}{p_\text{(photon)}} \) 5. Evaluate the result: Plug in the values computed in Steps 1 and 2 and the given force \(F = 0.100\,\text{N}\) to find the number of photons striking the surface per second.

Step-by-step solution

Find the energy of a single photon

Using the formula for the energy of a photon, we can calculate the energy of a single photon with a wavelength of 600 nm: \(E = \dfrac{hc}{\lambda}\), where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.63 \times 10^{-34}\ \text{Js}\)), \(c\) is the speed of light (\(3 \times 10^{8}\ \text{m/s}\)), and \(\lambda\) is the wavelength of the light (in meters). In our case, \(\lambda = 600 \times 10^{-9}\,\text{m}\).

Find the momentum of a single photon

Now, we can calculate the momentum of a single photon using the energy-momentum relation for photons: \(p = \dfrac{E}{c}\), where \(p\) is the momentum of the photon.

Calculate the force per photon

Since the photons are absorbed by the surface, the momentum transfer is equal to the initial momentum of the photons. Therefore, the force exerted by each photon is given by the momentum transfer rate: \(\Delta p_\text{(photon)} = \dfrac{dp}{dt} = p_\text{(photon)} \times \dfrac{dN}{dt} \), where \(\Delta p_\text{(photon)}\) is the momentum transfer rate, and \(\dfrac{dN}{dt}\) is the number of photons striking the surface per second.

Calculate the number of photons per second

Now, we can set the force exerted by the photons equal to the given force \(F = \Delta p_\text{(photon)} \) and solve for the number of photons per second: \(\dfrac{dN}{dt} = \dfrac{F}{p_\text{(photon)}} \) Plug in the values computed in Steps 1 and 2 and the given force \(F = 0.100\,\text{N}\) to find the number of photons striking the surface per second.

Evaluate the result

Using the formula derived in Step 4, evaluate the result for the number of photons per second needed to produce the given force on the surface.

Key concepts

Momentum of a Photon

Photons, though they are particles of light with no rest mass, carry momentum, which can be counterintuitive. The momentum of a photon is directly related to its energy and inversely related to its wavelength. The formula to find the momentum of a photon is simple:

\[\begin{equation} p = \frac{E}{c},\end{equation}\]
where \(p\) represents the photon's momentum, \(E\) is the energy, and \(c\) is the speed of light in a vacuum. This formula stems from the relativistic relationship between energy and momentum for massless particles like photons. It's important to note that since photons have momentum, they can exert a force upon interaction with matter, such as when they are absorbed or reflected, thus playing a crucial role in exercises like the one provided. Understanding this concept is key to solving problems involving light's particle-like behavior.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics that describes the quantization of energy. It is denoted by the symbol \(h\) and has a value of approximately \(6.63 \times 10^{-34}\) joule-seconds (Js). In the realm of photon energy, it can be used in tandem with the speed of light, represented as \(c\), to compute the energy of a photon given its wavelength \(\lambda\) using the equation:

\[\begin{equation} E = \frac{hc}{\lambda}.\end{equation}\]
Planck's constant essentially bridges the gap between the microscopic quantized world and the macroscopic world we observe, signifying that energy in quantum systems, such as the energy carried by photons, is not continuous but comes in discrete 'packets.' Recognizing its importance is paramount, as it helps us translate the physical properties of light into the energy and momentum values essential for calculations like the one in our exercise.

Energy-Momentum Relation

The energy-momentum relation is a crucial aspect of understanding how photons behave. This relationship reveals that the energy (\(E\)) of a photon is directly proportional to its momentum (\(p\)). Since photons move at the constant speed of light (\(c\)) in a vacuum, by algebraic manipulation, we derive the momentum as follows:

\[\begin{equation} p = \frac{E}{c}.\end{equation}\]
This relation implies that as the energy of a photon increases, so does its momentum. Consequently, higher frequency (or shorter wavelength) light exerts a greater force when interacting with matter compared to lower frequency (or longer wavelength) light, assuming the same number of photons are involved. In practical applications, especially in light-matter interactions such as in our exercise's scenario, this relation helps us calculate the overall force exerted by photons on a surface.

Photon Energy Calculation

Calculating the energy of a photon is a fundamental skill in understanding the interactions of light and matter. As mentioned earlier, the energy of a photon can be determined if its wavelength is known using the formula:

\[\begin{equation} E = \frac{hc}{\lambda},\end{equation}\]
where \(E\) is the energy, \(\lambda\) is the wavelength of the photon, \(h\) is Planck's constant, and \(c\) is the speed of light. By computing the energy of a photon, we are halfway through determining its momentum, which in turn, when multiplied by the number of photons striking a surface within a certain time, gives us the force exerted. The application of this photon energy calculation is demonstrated in the question posed, wherein by calculating the energy of a photon with a specific wavelength, it's possible to find the number of photons needed to achieve a given force on a surface.